User blog:B1mb0w/Beta Function Code Version 1
'Replaced by Version 2' This version has been replaced by Beta Function Version 2 'Beta Function - Sequence Generating Code' The Beta Function has been defined using program code shown below. A separate blog will be written to explain how Sequence Generator Code is compiled and executed using a normal programming language ... Work in Progress. 'Sequence Generating Code Version 1' The Beta Function is a significant re-design of the code used by the Alpha Function and will create long finite integer strings to define every large Veblen ordinal and FGH function for every finite integer (up to the size of \(f_{SV0}(n)\)). Refer to my other blogs on Unique Ordinal Representation and Program Code Version 4 for more background information. Beta Function Description *\(\beta(r,v) = h^m(v)\) where **\(h^0(v) = f_{\gamma_0}^{u_0}(v)\) where \(u_0\) and \(\gamma_0\) are in base \(v\), i.e. only use integers between \(0 ... v-1\) **\(h^{n + 1}(v) = f_{\gamma_{n + 1}}^{u_{n + 1}}(h^n(v))\) where ***\(u_{n + 1} < h^n(v)\) in base \(v\) ***\(\gamma_{n + 1} < \gamma_n\) in base \(h^n(v)\) i.e. only use integers between \(0 ... h^n(v) - 1\) **\(\gamma_m = 0\), i.e. the function recursively continues until we reach ordinal zero. and \(\beta(v^{v+1},v) = f_{\varphi(1,0_{v})}(v) = f_{SVO}(v)\) by definition The Beta Function can then be uniquely represented by a sequence of finite integers as follows: \(\beta(r,v) (v,h_0)\) The Sequence Generating RuleSet is as follows: *\(h_x = (d<2,d(0:x<1,P_h = 1)))\) *\(f_x = (g_x,g_x((0,0,0):h_u,(h_U,(f_{x+1} \alpha\) **\(0 < \gamma_e < \varphi(\beta,0)\uparrow\uparrow 2\) **\(0 < \gamma_c < \varphi(\beta,0)\uparrow\uparrow t\) **\(0 < \gamma_a < (\varphi(\beta,0)\uparrow\uparrow t)^{\gamma_e}.{\gamma_c}\) Using the following Real Number inputs into the Beta Function generates these ordinals: \(\beta(12.026783892,3) = f_{\varphi(1,\varphi(2,0) + 1)}(3)\) \(\beta(12.029332923,3) = f_{\varphi(1,\varphi(2,0).2)}(3)\) 'Test Bed for Version 1' Below is the test bed and various results using version 1. \(\beta(0,2) = 0 \) \(\beta(0.5,2) = 1 \) \(\beta(1,2) = 2 \) \(\beta(1.2,2) = 3 \) \(\beta(1.45,2) = 4 \) \(\beta(1.55,2) = 5 \) \(\beta(1.7,2) = 6 \) \(\beta(1.84,2) = 7 \) \(\beta(2.0001,2) = f_{\omega}(2) \) Second attempt - Base v = 2 - 20 Apr 2016 \(\beta(0,2) = 0\) \(\beta(0.5,2) = 1\) \(\beta(1,2) = 2\) \(\beta(1.2,2) = 3\) \(\beta(1.45,2) = 4\) \(\beta(1.55,2) = 5\) \(\beta(1.7,2) = 6\) \(\beta(1.84,2) = 7\) \(\beta(2.0001,2) = f_{\omega}(2)\) \(\beta(2.05,2) = f_{\omega}(2) + 1\) \(\beta(2.1,2) = f_{\omega}(2) + 2\) \(\beta(2.125,2) = f_{\omega}(2) + 3\) Next attempt - Base v = 2 - 21 Apr 2016 \(\beta(0,2) = 0 = 0\) \(\beta(0.5,2) = 1 = 1\) \(\beta(1,2) = 2 = 2\) \(\beta(1.2,2) = 3 = 3\) \(\beta(1.5,2) = 4 = 4\) \(\beta(1.6,2) = 5 = 5\) \(\beta(1.7,2) = 6 = 6\) \(\beta(1.9,2) = 7 = 7\) \(\beta(2.0001,2) = f_{\omega}(2) = 8\) \(\beta(2.05,2) = f_{\omega}(2) + 1 = 9\) \(\beta(2.1,2) = f_{\omega}(2) + 2 = 10\) \(\beta(2.125,2) = f_{\omega}(2) + 3 = 11\) \(\beta(2.135,2) = f_{\omega}(2) + 4 = 12\) \(\beta(2.15,2) = f_{\omega}(2) + 5 = 13\) \(\beta(2.16,2) = f_{\omega}(2) + 6 = 14\) \(\beta(2.17,2) = f_{\omega}(2) + 7 = 15\) \(\beta(2.2,2) = f_{\omega}(2) * 2 = 16\) \(\beta(2.21,2) = f_{\omega}(2) * 2 + 1 = 17\) Next attempt - Base v = 2 - 22 Apr 2016 \(\beta(0,2) = 0 = 0\) \(\beta(0.5,2) = 1 = 1\) \(\beta(1,2) = 2 = 2\) \(\beta(1.2,2) = 3 = 3\) \(\beta(1.5,2) = 4 = 4\) \(\beta(1.6,2) = 5 = 5\) \(\beta(1.7,2) = 6 = 6\) \(\beta(1.9,2) = 7 = 7\) \(\beta(2.0001,2) = f_{\omega}(2) = 8\) \(\beta(2.05,2) = f_{\omega}(2) + 1 = 9\) \(\beta(2.1,2) = f_{\omega}(2) + 2 = 10\) \(\beta(2.125,2) = f_{\omega}(2) + 3 = 11\) \(\beta(2.135,2) = f_{\omega}(2) + 4 = 12\) \(\beta(2.15,2) = f_{\omega}(2) + 5 = 13\) \(\beta(2.16,2) = f_{\omega}(2) + 6 = 14\) \(\beta(2.17,2) = f_{\omega}(2) + 7 = 15\) \(\beta(2.2,2) = f_{\omega}(2) * 2 = 16\) \(\beta(2.21,2) = f_{\omega}(2) * 2 + 1 = 17\) \(\beta(2.23,2) = f_{\omega}(2) * 2 + 2 = 18\) \(\beta(2.24,2) = f_{\omega}(2) * 2 + 3 = 19\) \(\beta(2.242,2) = f_{\omega}(2) * 2 + 4 = 20\) \(\beta(2.244,2) = f_{\omega}(2) * 2 + 5 = 21\) \(\beta(2.247,2) = f_{\omega}(2) * 2 + 6 = 22\) \(\beta(2.253,2) = f_{\omega}(2) * 2 + 7 = 23\) \(\beta(2.255,2) = f_{\omega}(2) * 2 + f_{\omega}(2) = 24\) \(\beta(2.28,2) = f_{1}^{2}(f_{\omega}(2))\) \(\beta(2.322,2) = f_{1}^{2}(f_{\omega}(2)) + f_{\omega}(2) * 2\) \(\beta(2.33,2) = f_{1}^{4}(f_{\omega}(2))\) \(\beta(2.3727,2) = f_{1}^{4}(f_{\omega}(2)) + f_{\omega}(2) * 2\) \(\beta(2.376,2) = f_{1}^{4}(f_{\omega}(2)) + f_{1}^{2}(f_{\omega}(2))\) \(\beta(2.3781,2) = f_{1}^{4}(f_{\omega}(2)) + f_{1}^{2}(f_{\omega}(2)) + f_{\omega}(2) * 2\) Second attempt - Base v = 3 - 22 Apr 2016 \(\beta(0,3) = 0\) \(\beta(0.5,3) = 1\) \(\beta(0.7,3) = 2\) \(\beta(1,3) = 3\) \(\beta(1.45,3) = 6\) \(\beta(1.75,3) = f_{1}^{2}(3)\) \(\beta(2.1,3) = f_{2}(3)\) \(\beta(2.28,3) = f_{2}(3) * 2\) \(\beta(2.34,3) = f_{1}^{2}(f_{2}(3))\) \(\beta(2.5,3) = f_{2}^{2}(3)\) \(\beta(3,3) = f_{\omega}(3)\) \(\beta(3.07,3) = f_{\omega}^{2}(3)\) \(\beta(3.141,3) = f_{\omega + 1}(3)\) \(\beta(3.214,3) = f_{\omega + 1}^{2}(3)\) \(\beta(3.224,3) = f_{1}^{f_{\omega}(3) + 2}(f_{\omega + 1}^{2}(3)) + f_{1}^{f_{6}^{f_{2}^{f_{2}^{2}(3) + 1}(f_{\omega}(3)) + f_{1}^{f_{2}(3) + 4}(f_{2}^{f_{1}^{2}(3)}(f_{\omega}(3))) + 3}(f_{\omega}^{2}(3))}(f_{f_{\omega}(3) + 1}^{2}(f_{\omega}^{2}(3)))\) \(\beta(3.288,3) = f_{\omega + 2}(3)\) Next attempt - Base v = 3 - 24 Apr 2016 \(\beta(0,3) = 0\) \(\beta(0.5,3) = 1\) \(\beta(0.7,3) = 2\) \(\beta(1,3) = 3\) \(\beta(1.45,3) = 6\) \(\beta(1.75,3) = f_{1}^{2}(3)\) \(\beta(2.1,3) = f_{2}(3)\) \(\beta(2.28,3) = f_{2}(3).2\) \(\beta(2.34,3) = f_{1}^{2}(f_{2}(3))\) \(\beta(2.5,3) = f_{2}^{2}(3)\) \(\beta(3,3) = f_{\omega}(3)\) \(\beta(3.07,3) = f_{\omega}^{2}(3)\) \(\beta(3.141,3) = f_{\omega + 1}(3)\) \(\beta(3.214,3) = f_{\omega + 1}^{2}(3)\) \(\beta(3.288,3) = f_{\omega + 2}(3)\) \(\beta(3.364,3) = f_{\omega + 2}^{2}(3)\) \(\beta(3.442,3) = f_{\omega.2}(3)\) \(\beta(3.9485,3) = f_{\omega^2}(3)\) \(\beta(5.1963,3) = f_{\omega\uparrow\uparrow 2}(3)\) \(\beta(5.197,3) = f_{\omega\uparrow\uparrow 2}(3) + 2\) \(\beta(5.2,3) = f_{1}^{6}(f_{\omega\uparrow\uparrow 2}(3)) + 1\) \(\beta(6,3) = f_{f_{1}^{2}(3) + 3}^{f_{f_{7}(f_{8}^{2}(f_{f_{1}^{2}(3)}(f_{\omega.2 + 1}(3)))) + 1}^{f_{\omega}(f_{\omega.2 + 1}(3))}(f_{\omega.2 + 1}^{2}(3))}(f_{\omega\uparrow\uparrow 2.(\omega) + \omega + 1}^{2}(3))\) \(\beta(7,3) = f_{\omega\uparrow\uparrow 2^2 + 2}(3) + f_{\omega.2 + 1}(f_{\omega\uparrow\uparrow 2^2}(3)) + f_{\omega^2 + \omega.(f_{\omega}^{f_{\omega}(3)}(f_{\omega^2 + 2}^{2}(3)))}(f_{\omega^2.2 + \omega.2}^{2}(3))\) \(\beta(8,3) = f_{\omega\uparrow\uparrow 2^2.2}^{f_{f_{1}^{2}(f_{2}^{2}(3)) + 2}(f_{\omega}(f_{\omega + 1}^{f_{\omega.2}(3) + 2}(f_{\omega.2}^{2}(3)))) + f_{\omega.2}^{2}(3)}(f_{\omega\uparrow\uparrow 2^2.(\omega + 1) + \omega^2.2 + 1}^{2}(3))\) \(\beta(8.5,3) = f_{1}^{3}(f_{\omega\uparrow\uparrow 2^2.(\omega^2 + 2)}(3)) + f_{\omega.(f_{3}^{f_{\omega}(f_{\omega.2 + 2}(3))}(f_{4}^{2}(f_{9}(f_{10}^{2}(f_{\omega^2 + 1}(3))))))}(f_{\omega^2 + \omega}(3))\) \(\beta(9,3) = f_{\varphi(1,0)}(3)\) \(\beta(9.00001,3) = f_{\varphi(1,0)}(3) + 1\) \(\beta(9.001434,3) = f_{\varphi(1,0) + 1}(3)\) \(\beta(9.004293,3) = f_{\varphi(1,0) + \omega}(3)\) \(\beta(9.0064395,3) = f_{\varphi(1,0) + \omega\uparrow\uparrow 2}(3)\) \(\beta(9.00859,3) = f_{\varphi(1,0).2}(3)\) \(\beta(9.0171825,3) = f_{\varphi(1,0).(\omega)}(3)\) \(\beta(9.021483,3) = f_{\varphi(1,0).(\omega^2)}(3)\) \(\beta(9.0257856,3) = f_{\varphi(1,0).(\omega\uparrow\uparrow 2)}(3)\) \(\beta(9.0344,3) = f_{\varphi(1,0)^2}(3)\) \(\beta(9.068926,3) = f_{\varphi(1,0)^{\omega}}(3)\) \(\beta(9.1035865,3) = f_{\varphi(1,0)^{\omega\uparrow\uparrow 2}}(3)\) Next attempt - Base v = 3 - 25 Apr 2016 \(\beta(0,3) = 0\) \(\beta(0.5,3) = 1\) \(\beta(0.7,3) = 2\) \(\beta(1,3) = 3\) \(\beta(1.45,3) = 6\) \(\beta(1.75,3) = f_{1}^{2}(3)\) \(\beta(2.1,3) = f_{2}(3)\) \(\beta(2.28,3) = f_{2}(3).2\) \(\beta(2.34,3) = f_{1}^{2}(f_{2}(3))\) \(\beta(2.5,3) = f_{2}^{2}(3)\) \(\beta(3,3) = f_{\omega}(3)\) \(\beta(3.07,3) = f_{\omega}^{2}(3)\) \(\beta(3.141,3) = f_{\omega + 1}(3)\) \(\beta(3.214,3) = f_{\omega + 1}^{2}(3)\) \(\beta(3.288,3) = f_{\omega + 2}(3)\) \(\beta(3.364,3) = f_{\omega + 2}^{2}(3)\) \(\beta(3.442,3) = f_{\omega.2}(3)\) \(\beta(3.9485,3) = f_{\omega^2}(3)\) \(\beta(5.1963,3) = f_{\omega\uparrow\uparrow 2}(3)\) \(\beta(5.197,3) = f_{\omega\uparrow\uparrow 2}(3) + 2\) \(\beta(5.2,3) = f_{1}^{4}(f_{\omega\uparrow\uparrow 2}(3)) + f_{\omega\uparrow\uparrow 2}(3)\) \(\beta(6,3) = f_{f_{1}^{2}(3) + 3}^{f_{f_{7}(f_{8}^{2}(f_{f_{1}^{2}(3)}(f_{\omega.2 + 1}(3)))) + 1}^{f_{\omega}(f_{\omega.2 + 1}(3))}(f_{\omega.2 + 1}^{2}(3))}(f_{\omega\uparrow\uparrow 2.(\omega) + \omega + 1}^{2}(3))\) \(\beta(7,3) = f_{\omega\uparrow\uparrow 2^2 + 2}(3) + f_{\omega.2 + 1}(f_{\omega\uparrow\uparrow 2^2}(3)) + f_{\omega^2 + \omega.(f_{\omega}^{f_{\omega}(3)}(f_{\omega^2 + 2}^{2}(3)))}(f_{\omega^2.2 + \omega.2}^{2}(3))\) \(\beta(8,3) = f_{\omega\uparrow\uparrow 2^2.2}^{f_{f_{1}^{2}(f_{2}^{2}(3)) + 2}(f_{\omega}(f_{\omega + 1}^{f_{\omega.2}(3) + 2}(f_{\omega.2}^{2}(3)))) + f_{\omega.2}^{2}(3)}(f_{\omega\uparrow\uparrow 2^2.(\omega + 1) + \omega^2.2 + 1}^{2}(3))\) \(\beta(8.5,3) = f_{1}^{4}(f_{\omega\uparrow\uparrow 2^2.(\omega^2 + 2)}(3)) + 1\) \(\beta(9,3) = f_{\varphi(1,0)}(3)\) \(\beta(9.00001,3) = f_{\varphi(1,0)}(3) + 1\) \(\beta(9.001435,3) = f_{\varphi(1,0) + 1}(3)\) \(\beta(9.0042933,3) = f_{\varphi(1,0) + \omega}(3)\) \(\beta(9.0064395,3) = f_{\varphi(1,0) + \omega\uparrow\uparrow 2}(3)\) \(\beta(9.0085872,3) = f_{\varphi(1,0).2}(3)\) \(\beta(9.0171825,3) = f_{\varphi(1,0).(\omega)}(3)\) \(\beta(9.021483,3) = f_{\varphi(1,0).(\omega^2)}(3)\) \(\beta(9.0257856,3) = f_{\varphi(1,0).(\omega\uparrow\uparrow 2)}(3)\) \(\beta(9.0343973,3) = f_{\varphi(1,0)^2}(3)\) \(\beta(9.068926,3) = f_{\varphi(1,0)^{\omega}}(3)\) \(\beta(9.1035865,3) = f_{\varphi(1,0)^{\omega\uparrow\uparrow 2}}(3)\) \(\beta(9.13838,3) = f_{\varphi(1,0)\uparrow\uparrow 2}(3)\) \(\beta(9.278887,3) = f_{\varphi(1,1)}(3)\) \(\beta(9.421555,3) = f_{\varphi(1,1)\uparrow\uparrow 2}(3)\) \(\beta(9.566416,3) = f_{\varphi(1,2)}(3)\) \(\beta(9.8628543,3) = f_{\varphi(1,\omega)}(3)\) \(\beta(10.32482425,3) = f_{\varphi(1,\omega\uparrow\uparrow 2)}(3)\) \(\beta(10.8084326,3) = f_{\varphi(1,\varphi(1,0))}(3)\) \(\beta(10.89120821,3) = f_{\varphi(1,\varphi(1,0)\uparrow\uparrow 2)}(3)\) \(\beta(11.844667,3) = f_{\varphi(2,0)}(3)\) \(\beta(11.847177,3) = f_{\varphi(2,0).2}(3)\) \(\beta(11.8496876,3) = f_{\varphi(2,0).(\omega)}(3)\) \(\beta(11.859737,3) = f_{\varphi(2,0)^2}(3)\) \(\beta(11.935378,3) = f_{\varphi(2,0)\uparrow\uparrow 2}(3)\) \(\beta(12.026783892,3) = f_{\varphi(1,\varphi(2,0) + 1)}(3)\) \(\beta(12.029332923,3) = f_{\varphi(1,\varphi(2,0).2)}(3)\) \(\beta(12.211703,3) = f_{\varphi(2,1)}(3)\) \(\beta(12.9802463,3) = f_{\varphi(2,\omega)}(3)\) \(\beta(13.58823285,3) = f_{\varphi(2,\omega\uparrow\uparrow 2)}(3)\) \(\beta(14.2246972615,3) = f_{\varphi(2,\varphi(1,0))}(3)\) \(\beta(14.8909732862,3) = f_{\varphi(2,\varphi(2,0))}(3)\) \(\beta(14.9099198273,3) = f_{\varphi(2,\varphi(2,0)\uparrow\uparrow 2)}(3)\) \(\beta(15.5884574,3) = f_{\varphi(\omega,0)}(3)\) \(\beta(20,3) = f_{\varphi(\omega^2.2 + \omega + 1,\omega\uparrow\uparrow 2.(\omega + 2) + \omega^2.2 + \omega + 2)^2.(\varphi(1,\varphi(2,\omega^2 + 2)^2.2 + \omega)\uparrow\uparrow 2^{\varphi(2,\omega^2 + 2)^{\omega\uparrow\uparrow 2^2.(\omega.2 + 2)}.(\omega.2)})}(3)\) \(\beta(27,3) = f_{\varphi(1,0,0)}(3)\) \(\beta(40,3) = f_{\varphi(\omega + 1,\varphi(1,1,\varphi(1,\varphi(\omega\uparrow\uparrow 2.(\omega^2 + 2) + \omega.2 + 1,\omega\uparrow\uparrow 2^2.2 + \omega\uparrow\uparrow 2.(\omega.2 + 1) + 1)\uparrow\uparrow 2^{\varphi(1,0)^{\omega\uparrow\uparrow 2.2} + \varphi(1,0)^2})),0)}(3)\) \(\beta(70,3) = f_{\varphi(\varphi(\omega.2 + 1,\varphi(1,\varphi(\omega + 1,2)^{\omega^2 + \omega + 2}.(\omega^2.2 + 2) + \varphi(\omega,1)\uparrow\uparrow 2.(\omega\uparrow\uparrow 2^2 + 1)).2 + \varphi(2,\omega.2 + 1)\uparrow\uparrow 2^{\omega\uparrow\uparrow 2^2.2 + \omega.2}.(\varphi(1,\omega\uparrow\uparrow 2))),0,0)}(3)\) \(\beta(81,3) = f_{\varphi(1,0,0,0)}(3) = f_{SVO}(3)\) by definition Category:Blog posts